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Minimizing the Number of Tardy Jobs with Uniform Processing Times on Parallel Machines

Authors Klaus Heeger , Hendrik Molter

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Author Details

Klaus Heeger
  • Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Hendrik Molter
  • Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel

Funding

  • Heeger, Klaus: Supported by the ISF, grant No. 1070/20.
  • Molter, Hendrik: Supported by the ISF, grant nr. 1470/24 and by the European Union’s Horizon Europe research and innovation program under grant agreement 949707.

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Klaus Heeger and Hendrik Molter. Minimizing the Number of Tardy Jobs with Uniform Processing Times on Parallel Machines. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.47

Abstract

In this work, we study the computational (parameterized) complexity of P∣ r_j, p_j = p ∣∑ w_j U_j. Here, we are given m identical parallel machines and n jobs with equal processing time, each characterized by a release date, a due date, and a weight. The task is to find a feasible schedule, that is, an assignment of the jobs to starting times on machines, such that no job starts before its release date and no machine processes several jobs at the same time, that minimizes the weighted number of tardy jobs. A job is considered tardy if it finishes after its due date.
Our main contribution is showing that P∣r_j, p_j = p∣∑ U_j (the unweighted version of the problem) is NP-hard and W[2]-hard when parameterized by the number of machines. The former resolves an open problem in Note 2.1.19 by Kravchenko and Werner [Journal of Scheduling, 2011] and Open Problem 2 by Sgall [ESA, 2012], and the latter resolves Open Problem 7 by Mnich and van Bevern [Computers & Operations Research, 2018]. Furthermore, our result shows that the known XP-algorithm by Baptiste et al. [4OR, 2004] for P∣r_j, p_j = p∣∑ w_j U_j parameterized by the number of machines is optimal from a classification standpoint. 
On the algorithmic side, we provide alternative running time bounds for the above-mentioned known XP-algorithm. Our analysis shows that P∣r_j, p_j = p∣∑ w_j U_j is contained in XP when parameterized by the processing time, and that it is contained in FPT when parameterized by the combination of the number of machines and the processing time. Finally, we give an FPT-algorithm for P∣r_j, p_j = p∣∑ w_j U_j parameterized by the number of release dates or the number of due dates. With this work, we lay out the foundation for a systematic study of the parameterized complexity of P∣r_j, p_j = p∣∑ w_j U_j.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Discrete mathematics
  • Computing methodologies → Planning and scheduling
Keywords
  • Scheduling
  • Identical Parallel Machines
  • Weighted Number of Tardy Jobs
  • Uniform Processing Times
  • Release Dates
  • NP-hard Problems
  • Parameterized Complexity

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